Question

Find the differential equation of the all straight lines which are at $a$ constant distance $p$ from origin

Answer 1

Equation of a straight line which is at a fixed distance ${}^{\prime }{p}^{\prime }$ from the origin is

$x\cos \theta +y\sin \theta =p⟶\left(1\right)$

On differentiating wrt $x$ we get

$\cos \theta +\sin \theta \frac{dy}{dx}=0$

$\Rightarrow \sin \theta \frac{dy}{dx}=-\cos \theta ⟶\left(2\right)$

From $\left(1\right)$ we have

$x\cos \theta =-y\sin \theta +p$

$\Rightarrow x\cos \theta =-y\sin \theta +p$

$\Rightarrow -x\sin \theta \frac{dy}{dx}=-y\sin \theta +p$

$\Rightarrow x\sin \theta \frac{dy}{dx}=y\sin \theta -p⟶\left(3\right)$

Differentiating $\left(3\right)$ we get,

$x\frac{d^{2}y}{{dx}^2}\left(\sin \theta \right)+\sin \theta \frac{dy}{dx}=\frac{dy}{dx}\sin \theta$

$\Rightarrow x\frac{d^{2}y}{{dx}^2}\sin \theta =0$

$\Rightarrow x\frac{d^{2}y}{{dx}^2}=0$

$\therefore$ the differentia equation of a family of straight lines at a fixed distance $p$ from the origin is given by $x\frac{d^{2}y}{{dx}^2}=0$

and ${xy}^{\prime \prime }=0$